issues with probability a man has $4$ children, given that atleast one of whom is a girl.Find the probability that he has $3$ girls and $1$ boy.
MY TRY :
probability of girl=$1/4$ and probability of boy=$3/4$ (my doubt is here, are these two probabiity correct? because it is said that atleast one of whom is a girl , not exactly 1)
=> 4c3 * (1/4)^4 * 3/4
 A: I would imagine that the following assumptions hold: each child is either male or female; each child has the same chance of being male as of being female; and the sex of each child is independent of the sex of the other.
If at least one of the four children is a girl, then there are 15 equally-likely possibilities: BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG.  Of these, there are four equally-likely cases where he has 3 girls and 1 boy (BGGG, GBGG, GGBG, GGGB).  Thus, the probability that he has 3 girls and 1 boy given that he has at least 1 girl is 4/15.
A: You can also see it like this. Define the events


*

*$A$ = at least one child is a girl

*$B$ = three children are girls and one is a boy


There are 16 cases in the whole, of which $15$ satisfy $A$ and $4$ satisfy $B$. Furthermore $B \cap A = B$.
Thus $p(A)=\frac{15}{16}$ and $p(B)=\frac{4}{16}$, and $$p(B|A)=\frac{p(B \cap A )}{p(A)}=\frac{p(B)}{p(A)}=\frac{4}{15}.$$
A: I would condition on all the possible # of girls:
$P(3G & 1B)= 
P(2G & 1B among three children|#G=1)P(#G=1)+
+P(1G & 1B among two children| #G=2)P(#G=2) +
+P( 1B  the fourth children |#G=3 )P(#G=3) +
+P(-1G |#G=4)P(#G=4)$
the last term is zero because if he has four girls he can't have a boy
$P(#G=1)=1$
$P(#G=2)=Bin(n=3,k=1,p=1/2)$
$P(#G=3)=Bin(n=3,k=2,p=1/2)$
$P(2G & 1B among three children|#G=1)= Bin(n=3,k=2,p=1/2)$
$P(1G & 1B among two children| #G=2)= Bin(n=2,k=1,p=1/2)$
$P( 1B  the fourth children |#G=3 )=1/2$
A: The man has four children of which have have atleast one girl so he may have 1 girl and three boys or two girls and two boys or three girls and one boy or four girls. so the probability that he has 3 girls and one boy is 1/4.
